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Tuesday, June 30, 2015

An introduction to prisms was presented in the chapter "Elements of Solid Geometry" of this course (see topics "Cylindrical Surface" and "Prisms"). We recommend to refresh this information prior to continue with properties and characteristics of prisms presented in this lecture.

Parallelepiped is a prism with a parallelogram as its directrix.

Arguably, parallelepiped is the most frequently occurring type of prisms, especially, if it's a rectangular parallelepiped (see below).

Theorem 1All faces of a parallelepiped are parallelograms.

Side diagonal is a diagonal of a side parallelogram (like AB' or BC').Base diagonal is a diagonal of a base parallelogram (like AC or B'D').Space diagonal is a diagonal connecting vertices of opposite bases lying inside a parallelepiped (that is, AC', BD', CA' and DB').

Right parallelepipeds (a particular case of parallelepipeds) are parallelepipeds with all side edges (AA', BB', CC' and DD') perpendicular to bases. All side faces of right parallelepiped are rectangular since side edges are perpendicular to base edges.

Rectangular parallelepipeds or cuboids (a particular case of right parallelepipeds) are right parallelepipeds with rectangular bases. So, all faces of rectangular parallelepiped are rectangular and all edges are perpendicular to sides that are not parallel to them (for instance, AB⊥AA'D'D, B'C'⊥CC'D'D etc.).

Cubes (a particular case of rectangular parallelepipeds) are parallelepipeds with all square faces.

All parallelepipeds fall into category of hexahedrons because they have six faces.

Obviously, all opposite faces of a parallelepiped are congruent and parallel since they are parallelograms with sides correspondingly parallel and equal in length.

Theorem 2All space diagonals of a parallelepiped AC', BD', CA' and DB' intersect in one point and are divided by that point in half.

The following is a kind of three-dimensional equivalent of Pythagorean Theorem.

Theorem 3In a rectangular parallelepiped a square of a space diagonal equals to a sum of squares of three edges that share a vertex with this diagonal.

Rectangular parallelepiped is fully defined by the length of any three edges sharing the same vertex (for instance, AB, AD and AA'), because all other edges are equal to one of them and all angles are right angles.

The area of each face of a rectangular parallelepiped is the area of a rectangle, that is a product of the length of two of its sides sharing the same vertex.If three edges sharing a vertex A are a, b and c, the combined area of all six faces of a rectangular parallelepiped equals to2ab+2bc+2ac=2(ab+bc+ac)

Friday, June 26, 2015

Below are a few construction problems.Recall certain principles of construction of planes in three-dimensional space that we have agreed upon:The plane is considered as completely defined and uniquely constructed if(a) there are three points not on the same line that are known to belong to a plane;(b) there is a line and a point outside of this line that are known to belong to a plane;(c) there are two intersecting lines that are known to belong to a plane;(d) there are two parallel lines that are known to belong to a plane.If two non-parallel planes are constructed, we assume that their intersection is fully defined and constructed.If a plane and a non-parallel to it line are constructed, we assume that their intersection is fully defined and constructed.

In the previous lectures we discussed some elementary construction problems like constructing a line or a plane in three-dimensional space parallel or perpendicular to a given line or plane passing through a given point or line. We recommend to refresh that material prior to addressing the problems below.

Problem 1Construct a plane that passes through a given point M and is parallel to two given skew lines a and b.?γ: γ∋M, γ ∥ a, γ ∥ b

Problem 2Construct a line that passes through a given point M, intersects a given line a and is parallel to a given plane γ.?b: b∋M, b∩a≠∅, b ∥ γ

Problem 3Construct a line that intersects two given lines a and b and is parallel to the third line c.?h: h∩a≠∅; h∩b≠∅; h ∥ c

Problem 4Construct a line that passes through a given point H and is perpendicular (not necessarily intersecting) to two given lines a and b.?h: h∋H; h⊥a; h⊥b

Problem 5Construct a plane that intersects a given plane γ at a given angle ∠φ and contains a given line a that is parallel to plane γ.a∩γ=∅; a ∥ γ;?δ: δ∋a; ∠(δ,γ)=∠φ

Problem 6Given a plane γ and two points A and B outside of this plane and lying in the same half-space (considering plane γ as a border between two half-spaces).Find a point M on plane γ such that a sum of lengths of segments AM+MB is minimal.A∉γ; B∉γ; AB∩γ=∅;?M: AM+MB = min.

Thursday, June 25, 2015

1. Prove that the distance from any point A on plane γ to plane δ, that is parallel to plane γ, is the same for any point A.γ ∥ δ;A∈γ; B∈δ; AB⊥δ;A'∈γ; B'∈δ; A'B'⊥δ;⇒ AB = A'B'

2. Given two parallel planes γ and δ and point A on plane γ.Prove that any line parallel to plane δ passing through point A lies completely within plane γ.γ ∥ δ; A∈γ; a ∥ δ; A∈ a⇒ a∈γ

3. Given two intersecting planes γ and δ. Also given are line a on plane γ and line b on plane δ. These two lines, a and b are parallel to each other.Prove that these lines are also parallel to a line of intersection c of planes γ and δ.a∈γ; b∈δ; a ∥ b; c = γ∩δ⇒ a ∥ c

4. Given a plane γ, line a on it and line b outside it that is parallel to line a.Prove that any plane δ, that contains line b and not parallel to plane γ, intersects plane γ along line c parallel or coinciding with line a.a∈γ; b∉γ; a ∥ b;b∈δ; δ∩γ=c⇒ a ∥ c

5. Given two skew lines a and b. Plane γ contains line a and is parallel to line b. Plane δ contains line b and is parallel to line a.Prove that planes γ and δ are parallel to each other.a∩b=∅; a ∦ b;a∈γ; b ∥ γ; b∈δ; a ∥δ;⇒ γ ∥ δ

6. Given plane γ and line a both perpendicular to line b.Prove that these plane and line are parallel to each other.b⊥γ; b⊥a;⇒ a ∥ γ

Tuesday, June 23, 2015

Axiom 1. If two points of a straight line belong to a plane, every point of this line belongs to this plane.

Axiom 2. If two planes have a common point, they intersect along a straight line passing through this point.

Axiom 3. For any three points not lying on the same straight line there is one and only one plane that contains them.

Now we are ready to solve problems.

1. Construct a plane that intersects three different planes.

2. Prove that two angles in three-dimensional space with correspondingly parallel sides, one formed by lines a and b intersecting at point P and another formed by lines a' and b' intersecting at point P', are congruent.P = a∩b; P' = a'∩b';a ∥ a'; b ∥ b'⇒ ∠aPb = a'P'b'

3. Prove that a line a parallel to another line b, that is perpendicular to a plane γ, is itself perpendicular to the same plane.a ∥ b; b⊥γ⇒ a⊥γ

4. Prove that two perpendiculars a and b to the same plane γ are parallel to each other.a⊥γ; b⊥γ⇒ a ∥ b

5. Given a plane γ and a straight line a parallel to it.Prove that the distance from any two points M and N on a line a to a plane γ is the same.a ∥ γ; M∈a; N∈a;P∈γ; MP⊥γ;Q∈γ; NQ⊥γ⇒ MP=NQ

6. Prove that two planes γ and δ, parallel to another plane ρ, are parallel to each other.γ ∥ ρ; δ ∥ ρ⇒ γ ∥ δ

Monday, June 22, 2015

Prior to discussing an angle between a line and a plane we have to define a very important concept of projection onto a plane.

Definition 1If there is a plane γ and a point P outside this plane, a projection of point P onto plane γ is a point on plane γ which is a base of a perpendicular dropped from point P onto this plane.If point P belongs to plane γ, its projection onto plane γ is this point itself.

Definition 2If there is a plane γ and any curve c in the space, a projection of curve c onto plane γ is a set of projections of every individual point of this curve c onto plane γ.

Theorem 1Projection of any straight line c onto any plane γ is a straight line on plane γ.

Note that a line and its projection onto any plane either intersect (if a line intersects a plane, since projection of a point of their intersection is this point itself) or are parallel (if the line is parallel to a plane).Therefore, a line and its projection on any plane always belong to some plane, they are not skew lines.

Definition 3An angle between a line and a plane is defined as an angle between this line and its projection onto this plane.

Theorem 2Consider plane γ and line c that intersect this plane at point M (M=c∩γ).Prove that an angle between line c and its projection onto plane γ (that is, an angle between line c and plane γ) is smaller than an angle between line c and any other line on plane γ that passes through point M and is not a projection of line c onto plane γ.

The purpose of this lecture is to introduce a concept of an angle between any two (even non-intersecting) lines in three-dimensional space.

First of all, we will consider angles between rays (half-lines) to define angles. Two rays in three-dimensional space with a common point of origin form an angle that we can consider from a two-dimensional viewpoint by constructing (one and only) plane that contains these two half-lines.

DefinitionWhatever the measure of an angle between two rays with a common origin in two-dimensional sense on the plane that contains them is - that, by definition, would be the measure of an angle they form in three-dimensional sense.

There is one nuance in this case. On the plane there are four angles between two rays with a common origin - one smaller, another that completes a full circle, and both with positive (counter-clockwise) or negative (clockwise) value in degrees or radians. Traditionally, to simplify the issue in the three-dimensional space, we consider only the smaller angle between two rays with a common origin and always measure it in positive units, so its value is always between zero and 1800 or π radians (inclusive on both sides).

Now we are ready to define the angle between two rays that do not share a common origin.

Consider we have two points in three dimensional space, A and B, and two rays, a and b, correspondingly originated from these points in some directions.Let's choose any point P in space (in can even coincide with A or B) and construct two rays originating in this point P: a' ∥ a and b' ∥ b.We assume that rays a' and b' are constructed not only parallel to, correspondingly, a and b, but also similarly directed, so, if we make a parallel shift from point P to point A, rays a' and a will coincide and, analogously, with shifting point P to point B in regards to rays b' and b.

A three-dimensional angle between rays a' and b' was defined above.By definition, this is the angle between rays a and b.

Thus, we have defined the angle between two rays in three-dimensional space that share or do not share a common origin. Let's discuss the correctness of this definition.

Our first task was to construct two rays, a' and b' originating from any point P and correspondingly parallel to rays a and b.This construction has been discussed in the previous lectures and is based on the fact that we can always construct one and only one plane that contains a straight line (a or b) and a point P, after which we can construct one and only one line in that plane through point P parallel to a line (a or b).So, construction is always possible and unique.

Our second task is to prove that, if we choose a different point Q in space as a new origin and construct rays a" and b" correspondingly parallel to rays a and b, we will end up with an angle between new rays a" and b" congruent to the one between rays a' and b'.Here is the proof:Ray a' is parallel to ray a, ray a" is parallel to ray a. Therefore, rays a' and a" are parallel to each other, as has been proven before. Similarly, b' ∥ b". Therefore, angles formed by pairs of rays, one - by a' and b', another - by a" and b", have correspondingly parallel sides. As has been proven before (see topic "Plane ∥ Plane" - Theorem 4), these angles are congruent.Therefore, the value of the angle between two rays in three-dimensional space is independent of the position of a new point of origin P that we used to define this angle, which proves that our definition makes sense.

Switching from rays to full lines presents no additional problems. There are four ways we can construct pairs of rays from two straight line. We define the smaller angle among these four combinations as the angle between the lines.

Now we can talk about perpendicularity of two lines, even if they do not intersect in three-dimensional space (skew lines).

Theorem 1Perpendicular to a plane is perpendicular to any line on that plane.

Theorem 2Perpendicular to a plane is perpendicular to any line parallel to that plane.

Theorem 3A plane, that is perpendicular to one of two skew lines perpendicular to each other, is parallel to another line.

Wednesday, June 17, 2015

Definitions1. A dihedral angle is called the right dihedral angle if the corresponding linear angle is the right angle (900 = π/2 radian).2. Two half-planes with a common edge are called perpendicular to each other if a dihedral angle they form is the right dihedral angle.3. Two intersecting planes are called perpendicular to each other if any one of four pairs of their half-planes with common edge along the line of intersection are perpendicular to each other.

There are four dihedral angles formed by four pairs of half-planes of two intersecting planes. Obviously, it's sufficient to have only one of these four to form the right dihedral angle for all four to be the same. This follows from the corresponding properties of linear angles - if one of two supplemental angles is the right angle, the other is the right angle as well.

Theorem 1Given a plane σ, a point A on it and a line p perpendicular to plane σ at point A.Prove that any plane that contains line p is perpendicular to plane σ.

The above theorem stated that a plane that contains a line perpendicular to another plane is itself perpendicular to that plane.So, from a perpendicular line we went to a perpendicular plane.We can go the other way around, from a perpendicular plane go to a perpendicular line.

Theorem 2Given two planes σ and τ perpendicular to each other (that is, the linear angle of any dihedral angle they form is the right angle.)Then there exists a line s contained in plane σ that is also perpendicular to plane τ.

Theorem 3Given two planes σ and τ perpendicular to each other (that is, the linear angle of any dihedral angle they form is the right angle.)Choose any point A on plane σ outside of the line of intersection d of these two planes and drop a perpendicular from this point A onto plane τ. Then this perpendicular is completely contained in plane σ.

Theorem 4Given two planes σ and τ intersecting along line p (σ∩τ = p).Another given plane γ is perpendicular to both planes σ and τ (σ⊥γ and τ⊥γ).Then the line of intersection p of planes σ and τ is perpendicular to plane γ.

Wednesday, June 3, 2015

We are considering three planes σ ∥ τ and γ that intersects both of them, correspondingly, at lines s and t.

Imagine that parallel planes σ and τ are positioned horizontally relatively to an observer, σ is above τ, and the intersecting plane γ is slanted in a way that an observer sees it from a side (so, all planes look like lines for an observer, two horizontal parallel to each other and one intersecting both of them).

Line s divides plane σ into left and right half-planes (we will use suffixes L and R).Line t divides plane τ into left and right half-planes (we will use suffixes L and R).Each of these lines, s and t, divide plane γ into up and down half-planes (we will use both subdivisions and suffixes U for up and D for down, which line s or t is used as an edge will be clear from the context).

Let's introduce some terminology similar to the one in plane geometry when we considered two parallel lines and another intersecting both of them.

Parallel Planes:These are planes σ and τ.

Transversal Plane:This is plane γ.

Vertical Dihedral Angles:∠σLsγU and ∠σRsγD∠σRsγU and ∠σLsγD∠τLtγU and ∠τRtγD∠τRtγU and ∠τLtγD

Corresponding Dihedral Angles:∠σLsγU and ∠τLtγU∠σRsγU and ∠τRtγU∠σLsγD and ∠τLtγD∠σRsγD and ∠τRtγD

Draw a plane δ perpendicular to edge s. It will be perpendicular to edge t as well because s ∥ t.All the dihedral angles now can be represented by corresponding linear angles. Since the above statements are true for linear angles, they are true for dihedral.

Tuesday, June 2, 2015

If we draw a straight line on a plane, it divides the plane into two half-planes. We will consider the points on that border line as being a part of either half-plane and will call it an edge of a half-plane.

Consider two intersecting planes σ and τ with line d being their intersection. This line d divides each plane in two half-planes:σ = σ1∪σ2 and τ = τ1∪τ2σ∩τ = d;σ1∩σ2 = d and τ1∩τ2 = d;

Definition: an object formed by two half-planes from two different planes that share the common edge is called a dihedral angle.Thus, in our case, σ1∪τ1 constitutes a dihedral angle.Other three dihedral angles formed by two intersecting planes σ and τ are:σ1∪τ2σ2∪τ1σ2∪τ2

Line d is an edge of a dihedral angle, half-planes forming the angle are called its faces.

A notation for a dihedral angle that includes the names of its two faces and an edge in between similar to this: ∠σ1dτ1 can be used (but rarely).

Consider a dihedral angle ∠σ1dτ1 and a plane γ intersecting its edge d and faces σ1 and τ1 perpendicularly to the edge d.This plane γ intersects half-plane σ1 by ray s and intersects half-plane τ1 by ray t. These two rays, s and t have a common vertex at point A of intersection of plane γ with the edge of our dihedral angle d.Obviously, since s∈γ, t∈ γ and d⊥γ, we can state that d⊥s and d⊥t.Angle formed by two rays s and t with a common vertex A is called a linear angle of a dihedral angle ∠σ1dτ1.If, instead of plane γ, we choose any other plane δ perpendicular to edge d, the other two rays formed by the intersection of plane δ with faces of our dihedral angle will form another linear angle congruent to the one produced by plane γ because corresponding rays will be parallel. Quick proof is based on two theorems we have already covered: two planes perpendicular to the same line are parallel to each other and, if two parallel planes are intersected by a third one, the lines of intersection are parallel.

So, as we see, any dihedral angle determines its corresponding linear angle. To obtain this linear angle we just have to construct a plane perpendicularly to the edge.The reciprocal statement is true as well. If linear angle is given, we can construct one and only one dihedral angle by performing the following:1. Construct one and only one perpendicular to a plane where two forming rays of our linear angle are lying through its vertex. This will be the edge of a dihedral angle.2. Construct one and only one first face of a dihedral angle through one ray of a linear angle and an edge constructed in the previous step.3. Construct one and only one second face of a dihedral angle through another ray of a linear angle and the same edge.

Both procedures, constructing a linear angle from a given dihedral angle and, opposite, constructing a dihedral angle from a given linear angle, involved steps where one and only one element can be constructed. This leads us to a statement that we can considered as proven by this uniqueness of construction elements: congruent dihedral angles have congruent corresponding linear angles and, vice versa, if corresponding linear angles of two dihedral angles are congruent, dihedral angles are congruent as well.

As an immediate consequence of the above correspondence between dihedral and linear angles, we can establish a measure (degrees or radians) of dihedral angles as a measure of corresponding linear angles. We can talk about two half-planes with common edge forming an acute or obtuse angle or being perpendicular to each other based on corresponding properties of their linear angles.

In plane geometry two rays a and b with common vertex form, in theory, two angles, sum of which equals to 3600. Moreover, if we take into consideration the direction (clockwise or counterclockwise), each of these two angles can be positive or negative, thus making four different numeric values.Traditionally, dihedral angles are considered in a simplified manner. Indeed, there are two different dihedral angles formed by two half-planes with common edge, but we usually consider only the smaller one that correspond the smaller linear angle and always measure it in positive units. Thus, we will only be dealing with dihedral angles measured from 0 to 180 degrees (or from 0 to π radians).

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